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| Mirrors > Home > MPE Home > Th. List > hlcomd | Structured version Visualization version GIF version | ||
| Description: The half-line relation commutes. Theorem 6.6 of [Schwabhauser] p. 44. (Contributed by Thierry Arnoux, 21-Feb-2020.) |
| Ref | Expression |
|---|---|
| ishlg.p | ⊢ 𝑃 = (Base‘𝐺) |
| ishlg.i | ⊢ 𝐼 = (Itv‘𝐺) |
| ishlg.k | ⊢ 𝐾 = (hlG‘𝐺) |
| ishlg.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
| ishlg.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
| ishlg.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
| ishlg.g | ⊢ (𝜑 → 𝐺 ∈ 𝑉) |
| hlcomd.1 | ⊢ (𝜑 → 𝐴(𝐾‘𝐶)𝐵) |
| Ref | Expression |
|---|---|
| hlcomd | ⊢ (𝜑 → 𝐵(𝐾‘𝐶)𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hlcomd.1 | . 2 ⊢ (𝜑 → 𝐴(𝐾‘𝐶)𝐵) | |
| 2 | ishlg.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
| 3 | ishlg.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
| 4 | ishlg.k | . . 3 ⊢ 𝐾 = (hlG‘𝐺) | |
| 5 | ishlg.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
| 6 | ishlg.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
| 7 | ishlg.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
| 8 | ishlg.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
| 9 | 2, 3, 4, 5, 6, 7, 8 | hlcomb 28611 | . 2 ⊢ (𝜑 → (𝐴(𝐾‘𝐶)𝐵 ↔ 𝐵(𝐾‘𝐶)𝐴)) |
| 10 | 1, 9 | mpbid 232 | 1 ⊢ (𝜑 → 𝐵(𝐾‘𝐶)𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 Itvcitv 28441 hlGchlg 28608 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-hlg 28609 |
| This theorem is referenced by: hlcgreulem 28625 opphllem4 28758 opphllem5 28759 opphl 28762 hlpasch 28764 lnopp2hpgb 28771 colhp 28778 cgrahl1 28824 cgrahl2 28825 cgrahl 28835 cgracol 28836 dfcgra2 28838 sacgr 28839 acopy 28841 acopyeu 28842 inaghl 28853 tgasa1 28866 |
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